A Divide-and-Conquer Algorithm for Distributed Optimization on Networks

TL;DR

This paper introduces a divide-and-conquer distributed optimization algorithm for networked systems, achieving exponential convergence and near-linear computational cost, outperforming existing methods in numerical tests.

Contribution

It presents a novel divide-and-conquer approach for decentralized optimization with exponential convergence and efficient implementation at fusion centers.

Findings

Exponential convergence of the algorithm.

Near-linear computational complexity.

Superior performance in numerical experiments.

Abstract

In this paper, we consider networks with topologies described by some connected undirected graph ${\mathcal{G}}=(V, E)$ and with some agents (fusion centers) equipped with processing power and local peer-to-peer communication, and optimization problem $\min_{{\boldsymbol x}}\big\{F({\boldsymbol x})=\sum_{i\in V}f_i({\boldsymbol x})\big\}$ with local objective functions $f_i$ depending only on neighboring variables of the vertex $i\in V$. We introduce a divide-and-conquer algorithm to solve the above optimization problem in a distributed and decentralized manner. The proposed divide-and-conquer algorithm has exponential convergence, its computational cost is almost linear with respect to the size of the network, and it can be fully implemented at fusion centers of the network. Our numerical demonstrations also indicate that the proposed divide-and-conquer algorithm has superior performance than popular decentralized optimization methods do for the least squares problem with/without $\ell^1$ penalty.